Find the transformed equation of the hyperbola xy = 4 when rotated 45 degrees.

The given hyperbola is 
xy = 4

The transformation matrix when the hyperbola is rotated by an angle of θ is


Note that for θ = 45°, cosθ = sinθ = 1/√2.
Therefore in the transformed coordinate system,


That is,


The graphs of the original and the rotated hyperbola are shown in the graph below.

(6,1)(5,4)slope = (4 - 1) / (5 - 6) = 3/-1 = -3y = mx + bslope(m) = -3use either of ur , = 6 and y = 1now sub and find b, the y int1 = -3(6) + b1 = -18 + b1 + 18 = b19 = bso ur equation is : y = -3x + we need it in standard formy = -3x + 3x to both sides3x + y = 19 < == standard form